In each iteration, we need to make two queries to the SFO. One for the base \(z_t\)  and the other for the shadow \(z_t^{\prime }\) . However, in the first step of \([\mathrm {SEG}]\) , we can use the gradient at the previous shadow \(z_{t-1}^{\prime }\)  so that we only have to make only one query in each iteration and remember the query's result of the previous step. This algorithm is called Optimistic Gradient or Popov's Extra-gradient [1]} which can be described as:
\(\begin{aligned}z_t^{\prime }&=z_t+\gamma _t\cdot V(z_{t-1}^{\prime };\omega _{t-1})\\z_{t+1}&=z_t+\gamma _t\cdot V(z_t^{\prime }; \omega _t).\end{aligned}[\mathrm {OG}]\) 
